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## GPS: Perceiving Variability in Graphs

Growing Problem Solvers provides four original, related, classroom-ready mathematical tasks, one for each grade band. Together, these tasks illustrate the trajectory of learners’ growth as problem solvers across their years of school mathematics.

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## Mathematical Explorations: A New Twist on Collaborative Learning

### classroom-ready activities

Research on students' learning has made it clear that learning happens through an interaction with others and through communication. In the classroom, the more students talk and discuss their ideas, the more they learn. However, within a one-hour period, it is hard to give everyone an equal opportunity to talk and share their ideas. Organizing students in groups distributes classroom talk more widely and equitably (Cohen and Lotan 1997).

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## Using KenKen to Build Reasoning Skills

Through KenKen puzzles, students can explore parity, counting, subsets, and various problem-solving strategies.

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## Improving Student Reasoning in Geometry

Parallel geometry tasks with four levels of complexity involve students in writing and understanding proof.

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## Counterexamples as Starting Points for Reasoning and Sense Making

Asked to “fix” a false conjecture, students combine their reasoning and observations about absolute value inequalities, signed numbers, and distance to write true mathematical statements.

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## Thinking Like a Mathematician

The moves that mathematicians use to generate new questions can also be used by teachers and students to tie content together and spur exploration.

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## Perimeter and Beyond

A simple problem-solving exercise encourages teachers to “start small” to reveal how third graders understand multiple math concepts simultaneously.

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## Is There a “Best” Rectangle?

In trying to find the ideal dimensions of rectangular paper for folding origami, students explore various paper sizes, encountering basic number theory, geometry, and algebra along the way.

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## Teaching the Perpendicular Bisector: A Kinesthetic Approach

Through movement-a welcome change of pace-students explore the properties of the perpendicular bisector.

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## Problem Solvers: Problem: Counting is for the birds

Connecting mathematical thinking to the natural world can be as simple as looking up to the sky. Volunteer bird watchers around the world help scientists gather data about bird populations. Counting all the birds in a large flock is impossible, so reasonable estimates are made using techniques such as those described in this problem scenario. Scientists draw on these estimates to describe trends in the populations of certain species and to identify areas for further research.